Search Results (136)

The Krawtchouk and Hahn families are discrete orthogonal polynomials defined on the integer pixel grid, yet as polynomials, they are entire functions of a complex variable. Adopting this complex variable and complex-valued viewpoint, we develop an image enhancement framework that connects discrete orthogonal polynomial theory with geometric function theory. Two operators are introduced. The Krawtchouk operator exploits the binomial weight, for which the parameter p concentrates the basis around a selectable tonal level $x=pN$, producing a localized contrast enhancement steerable toward shadows, midtones, or highlights. The Hahn operator uses the two-parameter Hahn polynomials, the discrete analogue of the Jacobi family, in which $(\alpha ,\beta )$ give asymmetric control of dark and light bands. Each operator is the real restriction of a holomorphic near identity map $\mathfrak{F}\left(z\right)=z+{\sum }_k{c}_k{\tilde{\varphi }}_k\left(z\right)$, with the intensity entering the discrete basis through $x=NI$, realized as a monotone 256-entry lookup table. We prove a bounded deviation estimate $|F-\mathrm{id}|\le {\sum }_k\left|c_k\right|$ on the intensity segment $[0, 1]$ and a positive slope condition that, via the Noshiro–Warschawski criterion, is a univalence condition for the analytic transfer map, ruling out intensity order reversal and oscillatory folding. Because Hahn polynomials lose orthogonality at high order in naive arithmetic, we show that a three-term recurrence with log-gamma weights preserves orthonormality to within about ${10}^{-10}$ on the full 8-bit grid, where single precision computation fails, and that the same scheme remains at the double-precision roundoff level on 10-, 12-, and 16-bit grids ($N=1023,4095,65,535$). The operators cost $\mathcal{O}\left(mL\right)$ table construction plus one lookup per pixel (about 3 ms for a $1024\times 1024$ color image), and a histogram-based differential entropy criterion selects the focus parameters automatically. Experiments on imagery of fine art, wildlife, archaeology, and architecture show that the Krawtchouk operator yields stronger localized contrast compared to seven classical methods, while the Hahn operator attains higher PSNR/SSIM, both as deterministic fast slope-controlled transforms. A diffusion MRI example further demonstrates that matching the focus parameter to the tonal mass adapts the same operators to dark-dominated medical scan imagery. Full article

In this paper, we introduce new geometric properties of analytic functions by utilizing the fractional integral operator associated with the Mittag-Leffler function. Specifically, we establish several framework criteria under which strong differential subordination as well as superordination hold across the product domain $U\times \overline{U}$, wherein the coefficients are holomorphic functions in $U$. For each investigated relation, the corresponding best dominant and best subordinant are explicitly determined. Utilizing these foundational outcomes, we subsequently derive novel strong sandwich-type theorems that bridge these dual concepts. Full article

In this work, a closed isotropic universe with scalar and spinor fields is considered within the framework of the extended phase space approach. Unlike conventional canonical quantization that yields the Wheeler–DeWitt equation, our method allows us to derive the Schrödinger equation for the wave function of the Universe directly from the path integral constructed with the Faddeev–Popov effective action, including gauge-fixing and ghost terms. We employ a mixed representation in the path integral: a coordinate representation for gravitational variables (lapse function and scale factor) and ghost fields, together with a holomorphic representation for matter fields. Assuming a conformally coupled scalar field as a first step, we obtain exact solutions to the Schrödinger equation under specific gauge conditions. To address the problem of vacuum divergences, we introduce supersymmetric multiplets of scalar and spinor fields, which systematically cancel the divergences and yield a finite vacuum energy in the closed universe. The resulting finite vacuum energy is interpreted as a topological Casimir effect. Full article

In this paper, we introduce and investigate a new subclass of bi-univalent Bazilevič-type functions associated with the Gregory generating function. The proposed class is defined by means of subordination and depends on the parameters $\nu$ and $\alpha$, which provides a broader framework containing several previously studied subclasses as special cases. By using the coefficient comparison method together with standard estimates for functions with positive real part, we obtain upper bounds for the initial Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$. Furthermore, the Fekete–Szegö functional $|a_3-\xi a_2^2|$ is estimated for functions belonging to this class. Several special cases are also derived to demonstrate the connection between the present results and earlier known results related to Gregory-type bi-univalent functions. Full article

Let $u:({\mathbb{C}}^m,0)\to (\mathbb{C},0)$ and $v:({\mathbb{C}}^n,0)\to (\mathbb{C},0)$ be holomorphic function germs that have isolated critical points at the origin. It is known that $\mu (u+v)=\mu \left(u\right)·\mu \left(v\right)$ holds, where $\mu$ denotes the Milnor number. In this paper, we consider a formula similar to the above one. More precisely, for Morse functions $f:M\to \mathbb{R}$ and $g:N\to \mathbb{R}$, let $C(f+g)$ denote the fiber product of two copies of $f+g$. Two of the main results are as follows: Firstly, under a generic condition on the critical values of f and g, the equation $\chi \left(C\right(f+g\left)\right)=\chi \left(C\right(f\left)\right)·\chi \left(C\right(g\left)\right)$ holds. Here $\chi$ denotes the Euler characteristic. Secondly, under a certain condition, the converse of the first result is also true. Full article

Let $\mathcal{A}=\{p+Jq:p,q\in \mathbb{C},J^2=-1\}$ be the commutative bicomplex-type algebra in which J commutes with the scalar imaginary unit. A Cauchy–Riemann-type operator $\overline{D}$ is studied on domains in ${\mathbb{C}}^2$. In the active coordinates $\xi =z_1-i{z}_2$ and $\eta =z_1+i{z}_2$, the equation $\overline{D}f=0$ is diagonal in the idempotent basis: the $e_{+}$-component is holomorphic in $\xi$ with $\eta$ as the parameter, while the $e_{-}$-component is holomorphic in $\eta$ with $\xi$ as the parameter. The expression $e_{+}F\left(\xi \right)+e_{-}G\left(\eta \right)$ is the parameter-independent subcase. From this decomposition, one obtains a slice characterization, a criterion for separatedness, a comparison with ordinary holomorphic functions of two complex variables, active-variable Cauchy formulas and estimates, local series with parameter-dependent coefficients, reflection symmetry, and Hardy and Bergman kernel lifts on the separated Hilbert spaces. Full article

The proliferation of AI-generated content and personalized AI systems has sharpened two fundamental and related computational problems: the progressive erosion of authentic identity in AI-mediated representations, and the growing difficulty of distinguishing human-originated from AI-generated behavioral and textual streams. This paper proposes a rigorous computational framework in which digital identity is formalized as a holomorphic function of complex time T = (a + ib) ∈ ℂ, where the real component Re$\left(T\right)$ encodes chronological progression and the imaginary component Im$\left(T\right)$ spans a continuum from episodic memory ($Im\left(T\right) < 0$) through the present moment ($Im\left(T\right) = 0$) to prospective imagination ($Im\left(T\right) > 0$). We argue that holomorphicity—enforced via Cauchy–Riemann regularization during CTNN learning (Proposition 1)—provides a theoretically grounded encoding of identity coherence, and discuss its advantages over alternative mathematical choices, including Lipschitz continuity, ${\mathit{C}}^{\mathbf{\infty }}$ smoothness, piecewise analytic functions, and stochastic models. Under four explicit Assumptions 1–4 covering the Markovian structure and fixed context window of current LLM architectures, we establish via Lemmas 1 and 2 and Theorem 1 that AI-generated behavioral trajectories exhibit structural limitations in satisfying the Cauchy–Riemann conditions at temporal depths characteristic of human biographical memory—limitations that do not arise for human trajectories learned under CTNN regularization. Building on this result, we introduce the Human–AI Authenticity Discriminant (HAAD), a theoretically grounded classifier with a fully specified calibration algorithm and sensitivity analysis ($\mathit{\kappa }$ ΔAUROC ≤ 0.04 over ±30% perturbation). Five metrics—TCS, ISI, PAS, GAS, and HAAD—are derived analytically from the holomorphic structure. The algorithmic framework is instantiated on four real-world datasets: MovieLens 25M, the Pushshift Reddit corpus, the Stack Overflow Data Dump, and the LIAR dataset. On the LIAR benchmark, TDT-HAAD achieves AUROC = 0.82 (95% CI: [0.79, 0.85]), exceeding a RoBERTa-based LLM detector baseline (AUROC = 0.75, DeLong p < 0.01); an ablation study supports the structural contribution of each component. A credibility harvesting signature is detectable 45.3 ± 12.1 days before standard temporal models reach statistical significance. Full article

This paper extends Loomis’s classical spectral criterion for almost periodic functions to bounded sequences in discrete settings. We first establish a discrete Kadets-type theorem for bounded sequences by regarding almost periodic functions as continuous with respect to the Bohr metric. We then introduce three spectra for bounded sequences via the Carleman transform and prove their equivalence. Using the Beurling–Gelfand theorem, we derive a discrete Loomis-type theorem, providing a spectral criterion for almost periodicity of bounded sequences. Our results extend the continuous theory to the discrete case and offer new tools for analyzing almost periodicity in sequence spaces. Full article

This paper develops a structured analytical framework and a robust numerical methodology for the spectral stability of time-varying bilateral quaternion differential equations of the form $\dot{q}=A\left(t\right)q+qB\left(t\right)$. By systematically extending classical real matrix theory to non-commutative dynamical systems via exact isometric real representations, this study utilizes the Kronecker product of real adjoint matrices to rigorously elucidate the underlying tensor structure of the bilateral evolution operator. This tensor-based reformulation proves that the Floquet multipliers of the bilaterally coupled system can be strictly decoupled into the product of the spectra corresponding to the left and right unilateral subsystems. Second, a “Scalar-Vector Stability Separation Principle” based on logarithmic norms is proposed, demonstrating that the transient energy evolution of the system is governed exclusively by the Hermitian real parts of the coefficient matrices, remaining entirely independent of the anti-Hermitian imaginary parts (rotation terms). Furthermore, for constant-coefficient and slowly varying systems, the Riesz projection from holomorphic functional calculus is introduced to establish algebraic criteria for exponential dichotomies, thereby revealing a cubic scaling law that relates the robustness threshold to the spectral gap (${\epsilon }_0\propto {\beta }^3$). Numerically, a Quaternion Chebyshev Spectral Collocation Method (Q-CSCM) is embedded within this exact vectorization framework to ensure that the algebraic symmetries of the bilateral system are strictly preserved through the isomorphic mapping. By explicitly constructing the fully discrete Kronecker product matrix via the exact real vectorization isomorphism, discrete energy estimates are utilized to rigorously prove that the numerical scheme successfully inherits the intrinsic spectral accuracy of the Chebyshev approximation. Comprehensive numerical experiments demonstrate that, within the low-dimensional regime, this methodology exhibits substantial temporal approximation efficiency advantages and superior numerical robustness compared to an alternative Legendre spectral baseline, as well as traditional explicit and state-of-the-art implicit symplectic Runge–Kutta methods, particularly when solving stiff and critically stable problems such as nonlinear Riccati oscillators. Full article

In this paper, we study a generalization of the Graham–Kohr extension operator, ${\mathsf{\Psi }}_{n,\alpha ,\beta }^{\gamma }\left(f\right)$, which maps functions defined on the unit disk into holomorphic mappings in the unit ball ${\mathbb{B}}^n$. Using the theory of Loewner chains, we show that, under suitable conditions, this operator can be embedded as the first element of a Loewner chain while preserving geometric properties. In addition, for suitable choices of the parameters, we establish subordination relations among starlike functions. Full article

Motivated by the fact that the Fibonacci sequence is the simplest nontrivial second-order recurrence with a rational generating function, we develop a Fibonacci-weighted Hardy theory for bicomplex holomorphic functions. Starting from the coefficient norm ${\sum }_{n\ge 0}{\left|a_n\right|}^2/F_{n+1}$, we obtain a bicomplex Hilbert module whose reproducing kernel is governed by ${(1-t-t^2)}^{-1}$ and whose maximal disk of holomorphy is determined sharply by the nearest kernel singularity, giving the radius ${\rho }_F={\phi }^{-1/2}$ (the square-root inverse of the golden ratio $\phi$). The arithmetic recurrence makes several objects fully explicit: we derive closed formulas for the kernels through the idempotent decomposition of $\mathbb{BC}$, compute exact norms of the shift powers and a golden-ratio spectral radius, and package the local theory into a sheaf of Fibonacci-holomorphic germs that are compatible with the bicomplex idempotent splitting. We also treat $(p,q)$-Fibonacci weights, obtaining a one-parameter family of rational kernels ${(1-pt-q{t}^2)}^{-1}$ and corresponding operator bounds. In addition to providing a concrete bicomplex model within weighted Hardy theory, the resulting explicit kernels furnish benchmark examples for kernel-based interpolation and for the operator theory of unilateral weighted shifts. Full article

In this paper, we introduce a new logarithmic integral operator that unifies differentiation and fractional integration within the complex domain. The present work addresses this gap by applying the proposed operator to analytic functions represented by alternating power series. The method demonstrates that the coefficients can be reorganized in a controlled manner without affecting convergence or analytic behavior. Using this framework, we derive third-order differential subordination and superordination results, which naturally lead to corresponding sandwich-type results. The findings confirm that the introduced operator offers an effective analytical tool for studying distortion, growth, and mapping properties of analytic functions, with promising potential for future applications in fluid mechanics. Full article

Using the instanton partition function for five-dimensional $U\left(1\right)$ gauge theory with eight supercharges and a single adjoint massive hypermultiplet on the $\Omega$ background, we give explicit expression for non-perturbative corrections to the topological string theory in the holomorphic limit. It was argued that in this case the theory is compactified on the twisted affine line bundle over $\mathbb{C}\times T^2$. We perform calculations in two ways. First we modify the integration contour by adding poles responsible for non-perturbative physics in accordance with a recent proposal. Then, we compute the genus zero Gopakumar–Vafa invariants for our case and evaluate the non-perturbative corrections to the partition function. We check that both calculations give the same result. Full article

The q-calculus framework has emerged as a powerful tool in geometric function theory, enabling refined analysis of analytic and bi-univalent functions. Inspired by the versatility of the q-derivative operator, this paper introduces a new generalized subclass of bi-univalent functions defined via the q-derivative in combination with generalized bivariate Fibonacci polynomials, which have recently gained significant attention in mathematical research. For functions in this class, we establish bounds on the initial coefficients and provide estimates for the corresponding Fekete–Szegö functional. By appropriate specialization of parameters, our results recover several known findings and, importantly, produce bounds for new subclasses of bi-univalent functions not previously studied. This framework unifies earlier developments while extending the theory to novel, analytically meaningful classes. Full article